Simplify $\left(3x^2 - 3 + 9x^3\right) - \left(4x^3 - 2x^2 + 16\right$\]A. $x^3 - 5x^2 + 25$B. $-x^3 + X^2 - 25$C. $5x^3 + X^2 + 13$D. $5x^3 + 5x^2 - 19$

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Understanding the Problem

The given problem involves simplifying an algebraic expression by combining like terms. This requires careful analysis and application of mathematical rules to arrive at the final simplified expression.

Step 1: Identify Like Terms

To simplify the given expression, we need to identify like terms, which are terms that have the same variable raised to the same power. In this case, the like terms are the terms with the variable xx raised to the power of 2 and the terms with the variable xx raised to the power of 3.

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them by adding or subtracting their coefficients. The given expression is:

(3x2−3+9x3)−(4x3−2x2+16)\left(3x^2 - 3 + 9x^3\right) - \left(4x^3 - 2x^2 + 16\right)

We can start by combining the like terms:

3x2−3+9x3−4x3+2x2−163x^2 - 3 + 9x^3 - 4x^3 + 2x^2 - 16

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by combining the like terms further. We can combine the terms with the variable xx raised to the power of 2 and the terms with the variable xx raised to the power of 3.

3x2−3+9x3−4x3+2x2−163x^2 - 3 + 9x^3 - 4x^3 + 2x^2 - 16

=(3x2+2x2)+(9x3−4x3)−3−16= (3x^2 + 2x^2) + (9x^3 - 4x^3) - 3 - 16

=5x2+5x3−19= 5x^2 + 5x^3 - 19

Step 4: Write the Final Answer

The final simplified expression is:

5x3+5x2−195x^3 + 5x^2 - 19

This matches option D.

Conclusion

In this article, we have simplified the given algebraic expression by combining like terms. We have identified the like terms, combined them, and simplified the expression to arrive at the final answer. The final answer is 5x3+5x2−195x^3 + 5x^2 - 19, which matches option D.

Final Answer

The final answer is 5x3+5x2−19\boxed{5x^3 + 5x^2 - 19}.

Discussion

This problem requires careful analysis and application of mathematical rules to arrive at the final simplified expression. The key concept in this problem is identifying like terms and combining them to simplify the expression.

Related Problems

This problem is related to other algebraic expression simplification problems. Some examples of related problems include:

  • Simplifying the expression (2x2+3x−1)+(x2−2x+4)\left(2x^2 + 3x - 1\right) + \left(x^2 - 2x + 4\right)
  • Simplifying the expression (4x3−2x2+1)−(3x3+x2−2)\left(4x^3 - 2x^2 + 1\right) - \left(3x^3 + x^2 - 2\right)

Tips and Tricks

Here are some tips and tricks for simplifying algebraic expressions:

  • Identify like terms and combine them to simplify the expression.
  • Use the distributive property to expand expressions.
  • Use the commutative and associative properties to rearrange expressions.
  • Use the order of operations to evaluate expressions.

Common Mistakes

Here are some common mistakes to avoid when simplifying algebraic expressions:

  • Failing to identify like terms and combine them.
  • Failing to use the distributive property to expand expressions.
  • Failing to use the commutative and associative properties to rearrange expressions.
  • Failing to use the order of operations to evaluate expressions.

Practice Problems

Here are some practice problems to help you practice simplifying algebraic expressions:

  • Simplify the expression (2x2+3x−1)+(x2−2x+4)\left(2x^2 + 3x - 1\right) + \left(x^2 - 2x + 4\right)
  • Simplify the expression (4x3−2x2+1)−(3x3+x2−2)\left(4x^3 - 2x^2 + 1\right) - \left(3x^3 + x^2 - 2\right)

Conclusion

In this article, we have simplified the given algebraic expression by combining like terms. We have identified the like terms, combined them, and simplified the expression to arrive at the final answer. The final answer is 5x3+5x2−195x^3 + 5x^2 - 19, which matches option D.

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Frequently Asked Questions

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify like terms, which are terms that have the same variable raised to the same power.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable raised to the same power. For example, in the expression 2x2+3x22x^2 + 3x^2, the terms 2x22x^2 and 3x23x^2 are like terms because they both have the variable xx raised to the power of 2.

Q: What is the next step after identifying like terms?

A: After identifying like terms, the next step is to combine them by adding or subtracting their coefficients.

Q: How do I combine like terms?

A: To combine like terms, add or subtract their coefficients. For example, in the expression 2x2+3x22x^2 + 3x^2, the coefficients are 2 and 3, so we add them to get 5x25x^2.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to simplify the expression by combining like terms further and rearranging the terms in the correct order.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to make the expression easier to work with and understand. It also helps to avoid errors and make calculations more efficient.

Q: Can you provide an example of simplifying an algebraic expression?

A: Here is an example of simplifying an algebraic expression:

(2x2+3x−1)+(x2−2x+4)\left(2x^2 + 3x - 1\right) + \left(x^2 - 2x + 4\right)

To simplify this expression, we first identify the like terms:

2x22x^2 and x2x^2 are like terms 3x3x and −2x-2x are like terms −1-1 and 44 are not like terms

Next, we combine the like terms:

2x2+x2=3x22x^2 + x^2 = 3x^2 3x−2x=x3x - 2x = x −1+4=3-1 + 4 = 3

So, the simplified expression is:

3x2+x+33x^2 + x + 3

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Failing to identify like terms and combine them
  • Failing to use the distributive property to expand expressions
  • Failing to use the commutative and associative properties to rearrange expressions
  • Failing to use the order of operations to evaluate expressions

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions on your own and checking your work with a calculator or online tool.

Conclusion

In this article, we have answered some frequently asked questions about simplifying algebraic expressions. We have covered the steps involved in simplifying an algebraic expression, including identifying like terms, combining them, and simplifying the expression further. We have also discussed the importance of simplifying algebraic expressions and provided some examples and tips for practicing simplification.